Dynamics of Flexible Multibody Systems

2. The tedious calculation of derivatives.

3. The geometrical description of the system.

4. The solution of the governing equation.

In order to avoid these difficulties many methods were developed. Kane developed in the early 1960s a new principle that has got more recognition in its application with multibody systems and it was called Lagrange‟s form of d‟Alembert‟s principle or Principle of virtual power. This principle states that the sum of the generalized active and inertia forces for each generalized coordinate (or alternatively, each generalized speed) is zero. The benefit of this form is that it automatically eliminates the non-working internal constraint forces without introducing any tedious differentiations [Huston 1990].

Lagrange‟s equations can be stated in the form:

d

d _{r r} ^r

K K

t x x F

  

  

  r1,2,...,n (3.1)

2 1

( ⁱ)

N

P R i i

K m







^{v (3.2)}

Such that xr (r=1,2,...,n) are the generalized coordinates describing the system configuration and n is the number of degrees of freedom. F_r is the generalized active force associated with the coordinate x_r. K is the system kinetic energy where P_i is a particle of the system and m_i is its mass. ^Rv^Pi is the velocity of Pi in an inertial reference frame R, and N is the number of particles in the system.

Lagrange‟s form of d‟Alembert‟s principle for a generalized coordinate system x_r ( r=1,2, ... , n) is then

*0

 _r

r F

F , r1, 2,..., ,n (3.3)

where Fr*

is the generalized inertia force for the generalized coordinate r and is expressed in the form

r p rp

r a x h

F^*    , (3.4)

where a_rp and h₁ are

rp k krm kpm kmn krm krm kpn

a m v v I    , (3.5)

r k krm kpm p kmn krm kpn p lsm ksn krm kql kpn q p

h m v v x I   x e I    x x

, (3.6)

arp is called the generalized inertia coefficient,

h_r is called the generalized inertia force coefficient and it contains the Coriolis and centrifugal

m_k is the mass of the particle P_k,

vk and ωk are the velocity and the angular velocity of the particle k, respectively, Ikmn is the mass inertia dyadic,

e_lsm are the components of the permutation symbol, and the repeated indices indicate a sum over the range of the index (from 1 to 3).

The dynamical equation may be written as:

r p rpx f

a   where f_r is defined as f_r F_rh_r. (3.7)

Equations (3.3) and (3.7) are applicable for unconstrained multibody systems with six degrees of freedom at each joint (three translational and three rotational). For the case of systems with fewer degrees of freedom the same equations also apply. With the reduction of a translational degree of freedom, the generalized coordinate associated with this degree of freedom must be zero. If a rotational degree of freedom is reduced then the terms associated with this degree cancel out from the dynamical equations. In case the movement at a joint is known then the generalized coordinate associated with that movement becomes a known variable.

By referring to the floating frame of reference formulation the introduction of the flexible bodies motion into a rigid bodies dynamic system is done by considering small linear body deformations relative to a local reference frame on the flexible body, while that local reference frame is undergoing large, non-linear global motion with respect to the other bodies in the dynamic system. Figure 3.9 shows an example of a flexible body with a body fixed local reference frame at one end called B, a vector s representing the position of a point P on the body before the deformation, and a vector uP representing the deformation in the body at that point.

The global coordinate system is called G.

Figure 3.9: The position vector of a deformed point P on a flexible body relative to the body fixed local reference frame B and the global coordinate system G [Intec 2005]

The flexible bodies are usually discretised into a large number of finite elements using the finite element method. In this method the infinite number of DOF of the flexible body are represented in a large finite number of DOF of the finite elements. Since the number of DOF in a finite element model is very large (it is time consuming in simulation of dynamic load cases) it is desirable to represent the deformation of the flexible body into a reduced model and derive the

sP

x uP

B

G P

governing equation of motion of the flexible body for this reduced system. One of the well-known and often used reduction methods is the Guyan reduction method, which is explained here and used in this research. In the Guyan method, a set of master nodes in the finite element model of the flexible body is defined by the user and retained. The rest of the nodes (slave nodes) are removed by condensation. This method is also called static condensation since the stiffness properties are considered during the condensation but the inertias coupling of master and slave nodes are ignored.

Through Guyan reduction the large, sparse FEM mass and stiffness matrices are condensed down into small, dense pair of matrices, with respect to the master DOF. The deformation here is described by a modal representation with a comparatively small number of modal coordinates. The following equations and derivations are based on data from the following resources [Wallrapp 1999, Adams 2008, and Intec 2005].

Firstly the linear deformations of the nodes of the finite element model, u, are approximated in a linear combination of a small number of shape vectors (or mode shapes), ϕ, which is also called modal superposition:

1 M

i i i

q







u φ ,

(3.8)

where M is the number of mode shapes and q are the modal coordinates. This equation can be represented in matrix form as:

uΦq, (3.9)

where Φ is the modal matrix containing the mode shapes. q is the vector of the modal coordinates. The modal matrix Φ becomes after modal truncation a rectangular matrix which is the transformation matrix from the modal coordinates to the physical coordinate u.

Since the free selection of the modes can lead to accidental constraints in the system a technique called Craig-Bampton method is to be used. In this method the system DOF are divided into two groups: the first group is the boundary DOF. These DOF are not subject to modal superposition and are preserved exactly in the Craig-Bampton modal basis. In these modes there are no losses of resolution when higher order modes are reduce and truncated. The second group is the interior DOF. Also two sets of mode shapes are defined in this method: Constraint modes gained by giving each boundary DOF a unit displacement while holding all other boundary DOF fixed. These modes are static shapes and span all possible motions of the boundary DOFs. The second group contains the fixed boundary normal modes obtained by fixing the boundary DOF and computing the eigen solutions.

The physical DOF and the Craig-Bampton modes with their coordinates are illustrated in the following equation

C B

IC IN N

I

   

    

     

I 0 q u u

Φ Φ q

u , (3.10)

where,

u_B is the boundary DOF, u_I is the interior DOF,

I and 0 are the identity and zero matrices, respectively,

ΦIC is the physical displacement of the interior DOF in the constraint modes, Φ is the physical displacement of the interior DOF in the normal modes,

q_C are the modal coordinates of the constraint modes,

qN are the modal coordinates of the fixed-boundary normal modes.

By the use of modal transformation, the generalized mass and stiffness matrices corresponding to the Craig-Bampton modal basis are:

ˆ ˆ

ˆ

T

CC

BB BI

T

IC IN IB II IC IN NN

 

    

       

 

     

I 0 K K I 0 K 0

K Φ KΦ

Φ Φ K K Φ Φ 0 K

, (3.11)

ˆ ˆ

ˆ ˆ ˆ

T

CC NC

BB BI

T

IC IN IB II IC IN CN NN

 

    

       

 

     

I 0 M M I 0 M M

M Φ MΦ

Φ Φ M M Φ Φ M M

. (3.12)

The subscripts I, B, N, and C denote the internal DOF, boundary DOF, normal modes, and constraint modes, respectively. M^ˆ and K^ˆ are the generalized mass and stiffness matrices.

Now for the reduced system the eigenvalues can be calculated from the following equation for each modal coordinate q

ˆ ˆ

(KM q) 0. (3.13)

From Figure 3.9 the location of a point P on the flexible body at some point of time is the sum of the three vectors, x

is the translation of body fixed local reference frame B with respect to a global inertia frame G, s_P the position of the point P before deformation with respect to the local body fixed reference frame, and u_P is the translational deformation vector of the point P from its undeformed position to the new deformed position.

  

P P P

r x s u (3.14)

and in matrix form:

 

G B

P   P P

r x A s u , (3.15)

where the values s_P and u_P are expressed in the local body coordinate system. ^GA^B is the transformation matrix from the local body reference frame B to the ground. From Eq. (3.9) the deformation u_P is a modal superposition as in the following equation:

P  P

u Φ q. (3.16)

ΦP is a slice from the modal matrix that corresponds to the translational DOF of node P.

The generalized coordinates of the flexible body are:

,( 1... ) i i M

x y z

q









 

 

 

   

   

   

   

   

 

 

 

x

ξ ψ

q

. (3.17)

M is the number of modes and qi are the generalized coordinates of the flexible body.

In order to compute the governing equation of motion of the flexible body the following terms are required:

The velocity of the point P in the system

 

G B G B

P  P P  P

v x A s u Bψ A Φ q, where Bψ^G ω^B_B . (3.18)

Gω^BB is the angular velocity of the flexible body relative to the global coordinate system expressed in the body coordinates. This velocity is displayed in terms of the time derivative of the generalized coordinate vector ξ in the form:

 

[ ^{G B G B} ]

P  P P  P

v I A s u B A Φ ξ, (3.19)

and the angular velocity of a marker on a flexible body at the point P with respect to the global coordinate system is

*

G P G B B P G B

B  B  B  B P

ω ω ω ω Φ q, (3.20)

where Φ^* is the slice from the modal matrix that corresponds to the rotational DOF of node P.

Now that the velocity and angular velocity are calculated, the governing equations of motion of flexible bodies are derived from Lagrange‟s equations which are function of the kinetic energy and the potential energy. This derivation needs to calculate the following terms:

The kinetic energy is 1

2

T G B G B

P P P P P P

p

T 



m ^{v v}  ^{ω I ω , (3.21)}

which can be displayed in the generalized mass matrix and generalized coordinate system as

1 ( )

2

T  ξ M ξ ξT . (3.22)

The mass matrix ( )

tt tr tm

T

tr rr rm

T T

tm rm mm

 

 

  

 

 

M M M

M ξ M M M

M M M

, (3.23)

where the subscripts t, r, and m are for the translational, rotational and modal DOF, respectively. For further information on the elements of this mass matrix see [Wallrapp 1999], [Adams 2008], [Intec 2005].

The potential energy, which comes out of two sources (the gravity and the elasticity), is:

( ) 1 2

T

V Vg ξ  ξ Kξ , (3.24)

where K is the generalized stiffness matrix, and it is constant. Since the modal coordinates contribute to the elastic energy it can be reduced into the form

tt tr tm

T

tr rr rm

T T

tm rm mm mm

   

   

   

   

   

K K K 0 0 0

K K K K 0 0 0

K K K 0 0 K

. (3.25)

Kmm is the generalized stiffness matrix of the structural components with respect to the modal coordinates.

The gravitational energy is then

[ ( ( ) ]^T d

g P

V

V 



 ^x^{A s} ^Φ P ^{q g} V^{, (3.26)}

and the gravitational force is then calculated from the gravitational energy in the generalized coordinate system as

d

( ) ( ( ) ) d

( ) d

V

T

g T

g P

V

T T

V

V

V P q V

P V







   

   

   

  

   

      

   

   

   

 







g

ξ A

f s Φ g

ξ ψ

Φ A g

. (3.27)

The damping in the system is represented in the form of damping force through Rayleigh‟s dissipation function, where D is the modal damping matrix. This function contains the damping coefficients and is generally constant

1 2

 T

F q Dq. (3.28)

Now the final form of the governing differential equation of motion of a flexible body in the generalized coordinates is

1 2

T T

g 

 

   

           

M Ψ

Mξ Mξ ξ ξ Kξ f Dξ Q

ξ ξ , (3.29)

where λ are Lagrange multipliers of the constraints, Ψ are the algebraic constraint equations, and Q are the generalized applied forces.

Im Dokument Advanced Simulation of an Adaptive Lower Limb Prosthesis (Seite 39-46)